Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0207186 · Advances In Physics

d x 2 - y 2 superconductivity and the Hubbard model

N. Bulut

Published Nov 1, 2002• 88 citations
arXivHTMLPDF

Abstract

The numerical studies of d x 2 - y 2 -wave pairing in the two-dimensional (2D) and the 2-leg Hubbard models are reviewed. For this purpose, the results obtained from the determinantal Quantum Monte Carlo and the Density-Matrix Renormalization-Group calculations are presented. These are calculations which were motivated by the discovery of the high- T c cuprates. In this review, the emphasis is placed on the microscopic many-body processes which are responsible for the d x 2 - y 2 -wave pairing correlations observed in the 2D and the 2-leg Hubbard models. In order to gain insight into these processes, the results on the effective pairing interaction as well as the magnetic, density and the single-particle excitations will be reviewed. In addition, comparisons will be made with the other numerical approaches to the Hubbard model and the numerical results on the t - J model. The results reviewed here indicate that an effective pairing interaction which is repulsive at ( ~ , ~ ) momentum transfer, and enhanced single-particle spectral weight near the ( ~ ,0) and (0, ~ ) points of the Brillouin zone, create optimum conditions for d x 2 - y 2 -wave pairing. These are two effects which act to enhance the d x 2 - y 2 -wave pairing correlations in the Hubbard model. Finding additional ways is an active research problem.

DOI
10.1080/00018730210155142

Related papers

The Pairing Interaction in the 2D Hubbard Model

Thomas Maier, Mark Jarrell, Douglas J. Scalapino

A dynamic cluster quantum Monte Carlo approximation is used to study the effective pairing interaction of a 2D Hubbard model with a near neighbor hopping $t$ and an on-site Coulomb interaction $U$ . The effective pairing interaction is characterized in terms of the momentum and frequency dependence of the eigenfunction of the leading eigenvalue of the irreducible particle-particle vertex. The momentum dependence of this eigenfunction is found to vary as $(\cos k_x-\cos k_y)$ over most of the Brillouin zone and its frequency dependence is determined by the exchange energy $J$. This implies that the effective pairing interaction is attractive for singlets formed between near-neighbor sites and retarded on a time scale set by $J^{-1}$. The strength of the pairing interaction measured by the size of the d-wave eigenvalue peaks for $U$ of order the bandwidth $8t$. It is found to increase as the system is underdoped.

arXiv:cond-mat/0606003•May 31, 2006•0 citations

Pairing correlations in the two-leg Hubbard ladder

Niyazi Bulut, Sadamichi Maekawa

In the ground state of the doped two-leg Hubbard ladder there are power-law decaying d_{x^2-y^2}-type pairing correlations. It is important to know the strength and the temperature scale of these correlations. For this purpose, we have performed determinantal Quantum Monte Carlo (QMC) calculations of the reducible particle-particle interaction in the Hubbard ladder. In this paper, we report on these calculations and show that, at sufficiently low temperatures, resonant particle-particle scattering takes place in the d_{x^2-y^2} pairing channel for certain values of the model parameters. The QMC data presented here indicate that the d_{x^2-y^2} pairing correlations are strong in the Hubbard ladder.

•Jul 30, 2005•0 citations

Enhancement of the dx2−y2 pairing correlation in the two-dimensional Hubbard model: A quantum Monte Carlo study

Kazuhiko Kuroki, Hideo Aoki

Quantum Monte Carlo is used to investigate the possibility of ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductivity in the two-dimensional repulsive Hubbard model. A small energy scale relevant to possible pairing requires a care [i.e., sufficiently small level separation between the $k$ points $(\ensuremath{\delta}k,\ensuremath{\pi}\ensuremath{-}\ensuremath{\delta}{k}^{\ensuremath{'}})$ and $(\ensuremath{\pi}\ensuremath{-}\ensuremath{\delta}{k}^{\ensuremath{'}\ensuremath{'}},\ensuremath{\delta}{k}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}})$ with small $\ensuremath{\delta}k$'s] to detect enhanced correlations in finite-size studies, as motivated from a previous study on Hubbard ladders. Our calculation indeed detects long-tailed enhancements in the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing correlation when the system is near, but not exactly at, half-filling.

arXiv:cond-mat/9707277•Physical review. B, Condensed matter•Dec 1, 1997•30 citations

Superconductivity in the two-dimensional Hubbard model

J. Beenen, David M. Edwards

Quasiparticle bands of the two-dimensional Hubbard model are calculated using the Roth two-pole approximation to the one-particle Green's function. Excellent agreement is obtained with recent Monte Carlo calculations, including an anomalous volume of the Fermi surface near half-filling, which can possibly be explained in terms of a breakdown of Fermi liquid theory. The calculated bands are very flat around the (\ensuremath{\pi},0) points of the Brillouin zone in agreement with photoemission measurements of cuprate superconductors. With doping there is a shift in spectral weight from the upper band to the lower band. The Roth method is extended to deal with superconductivity within a four-pole approximation allowing electron-hole mixing. It is shown that triplet p-wave pairing never occurs. A self-consistent solution with singlet ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is found and optimal doping occurs when the van Hove singularity, corresponding to the flat band part, lies at the Fermi level. Nearest-neighbor antiferromagnetic correlations play an important role in flattening the bands near the Fermi level and in favoring superconductivity. However, the mechanism for superconductivity is a local one, in contrast to spin-fluctuation exchange models. For reasonable values of the hopping parameter the transition temperature ${\mathit{T}}_{\mathit{c}}$ is in the range 10--100 K. The optimum doping ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$ lies between 0.14 and 0.25, depending on the ratio U/t. The gap equation has a BCS-like form and 2${\mathrm{\ensuremath{\Delta}}}_{\mathrm{max}}$/${\mathit{kT}}_{\mathit{c}}$\ensuremath{\simeq}4.

arXiv:cond-mat/9503054•Physical review. B, Condensed matter•Nov 1, 1995•111 citations

Low-Energy Electronic Excitations of the Layered Cuprates and the Hubbard Model

N. BULUT

We present a review of the Quantum Monte Carlo computation results on the magnetic, charge and single--particle excitations as well as the pairing correlations of the two--dimensional Hubbard model. We are particularly interested in how these quantities are related to each other in the metallic state that forms near half--filling. These results are useful in gaining a better understanding of the low energy excitations of the superconducting cuprates.

arXiv:cond-mat/9606204•TURKISH JOURNAL OF PHYSICS•Jan 1, 1996•5 citations

Enhanced dx2−y2 pairing correlations in the two-leg Hubbard ladder

R. M. Noack, N. Bulut, Douglas J. Scalapino +1 more

The two-leg Hubbard ladder is characterized by the ratio of the inter- to intra-leg hopping ${t}_{\ensuremath{\perp}}/t$, the relative interaction strength $U/t,$ and the electron filling. Here, using density matrix renormalization group and Monte Carlo simulations, we examine the dependence of the pairing correlations on these parameters. We find that the pairing correlations are enhanced when the top of the bonding quasiparticle band and the bottom of the antibonding band are near the Fermi level. We present results on the single-particle spectral weight and the antiferromagnetic correlations in order to explain this behavior.

arXiv:cond-mat/9612165•Physical review. B, Condensed matter•Sep 15, 1997•94 citations
The numerical studies of d x 2 - y 2 -wave pairing in the two-dimensional (2D) and the 2-leg Hubbard models are reviewed. For this purpose, the results obtained from the determinantal Quantum Monte Carlo and the Density-Matrix Renormalization-Group calculations are presented. These are calculations which were motivated by the discovery of the high- T c cuprates. In this review, the emphasis is placed on the microscopic many-body processes which are responsible for the d x 2 - y 2 -wave pairing correlations observed in the 2D and the 2-leg Hubbard models. In order to gain insight into these processes, the results on the effective pairing interaction as well as the magnetic, density and the single-particle excitations will be reviewed. In addition, comparisons will be made with the other numerical approaches to the Hubbard model and the numerical results on the t - J model. The results reviewed here indicate that an effective pairing interaction which is repulsive at ( ~ , ~ ) momentum transfer, and enhanced single-particle spectral weight near the ( ~ ,0) and (0, ~ ) points of the Brillouin zone, create optimum conditions for d x 2 - y 2 -wave pairing. These are two effects which act to enhance the d x 2 - y 2 -wave pairing correlations in the Hubbard model. Finding additional ways is an active research problem.
A dynamic cluster quantum Monte Carlo approximation is used to study the effective pairing interaction of a 2D Hubbard model with a near neighbor hopping ttt and an on-site Coulomb interaction UUU . The effective pairing interaction is characterized in terms of the momentum and frequency dependence of the eigenfunction of the leading eigenvalue of the irreducible particle-particle vertex. The momentum dependence of this eigenfunction is found to vary as (cos⁡kx−cos⁡ky)(\cos k_x-\cos k_y)(coskx​−cosky​) over most of the Brillouin zone and its frequency dependence is determined by the exchange energy JJJ. This implies that the effective pairing interaction is attractive for singlets formed between near-neighbor sites and retarded on a time scale set by J−1J^{-1}J−1. The strength of the pairing interaction measured by the size of the d-wave eigenvalue peaks for UUU of order the bandwidth 8t8t8t. It is found to increase as the system is underdoped.
In the ground state of the doped two-leg Hubbard ladder there are power-law decaying d_{x^2-y^2}-type pairing correlations. It is important to know the strength and the temperature scale of these correlations. For this purpose, we have performed determinantal Quantum Monte Carlo (QMC) calculations of the reducible particle-particle interaction in the Hubbard ladder. In this paper, we report on these calculations and show that, at sufficiently low temperatures, resonant particle-particle scattering takes place in the d_{x^2-y^2} pairing channel for certain values of the model parameters. The QMC data presented here indicate that the d_{x^2-y^2} pairing correlations are strong in the Hubbard ladder.
Quantum Monte Carlo is used to investigate the possibility of ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductivity in the two-dimensional repulsive Hubbard model. A small energy scale relevant to possible pairing requires a care [i.e., sufficiently small level separation between the kkk points $(\ensuremath{\delta}k,\ensuremath{\pi}\ensuremath{-}\ensuremath{\delta}{k}^{\ensuremath{'}})$ and $(\ensuremath{\pi}\ensuremath{-}\ensuremath{\delta}{k}^{\ensuremath{'}\ensuremath{'}},\ensuremath{\delta}{k}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}})$ with small $\ensuremath{\delta}k$'s] to detect enhanced correlations in finite-size studies, as motivated from a previous study on Hubbard ladders. Our calculation indeed detects long-tailed enhancements in the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing correlation when the system is near, but not exactly at, half-filling.
Quasiparticle bands of the two-dimensional Hubbard model are calculated using the Roth two-pole approximation to the one-particle Green's function. Excellent agreement is obtained with recent Monte Carlo calculations, including an anomalous volume of the Fermi surface near half-filling, which can possibly be explained in terms of a breakdown of Fermi liquid theory. The calculated bands are very flat around the (\ensuremath{\pi},0) points of the Brillouin zone in agreement with photoemission measurements of cuprate superconductors. With doping there is a shift in spectral weight from the upper band to the lower band. The Roth method is extended to deal with superconductivity within a four-pole approximation allowing electron-hole mixing. It is shown that triplet p-wave pairing never occurs. A self-consistent solution with singlet ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is found and optimal doping occurs when the van Hove singularity, corresponding to the flat band part, lies at the Fermi level. Nearest-neighbor antiferromagnetic correlations play an important role in flattening the bands near the Fermi level and in favoring superconductivity. However, the mechanism for superconductivity is a local one, in contrast to spin-fluctuation exchange models. For reasonable values of the hopping parameter the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ is in the range 10--100 K. The optimum doping ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$ lies between 0.14 and 0.25, depending on the ratio U/t. The gap equation has a BCS-like form and 2${\mathrm{\ensuremath{\Delta}}}_{\mathrm{max}}$/kTc{\mathit{kT}}_{\mathit{c}}kTc​\ensuremath{\simeq}4.
The two-leg Hubbard ladder is characterized by the ratio of the inter- to intra-leg hopping ${t}_{\ensuremath{\perp}}/t$, the relative interaction strength U/t,U/t,U/t, and the electron filling. Here, using density matrix renormalization group and Monte Carlo simulations, we examine the dependence of the pairing correlations on these parameters. We find that the pairing correlations are enhanced when the top of the bonding quasiparticle band and the bottom of the antibonding band are near the Fermi level. We present results on the single-particle spectral weight and the antiferromagnetic correlations in order to explain this behavior.